Jianliang Qian

• Michigan State University

Starting from Kirchhoff-Huygens representation and Duhamel’s principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard’s ansatz into the Kirchhoff-Huygens represen...

We propose to use the Kantorovich-Rubinstein (K-R) metric as a novel misfit function for the level-set based inverse gravity problems, where modulus of gravity-force data is used. By using the modulus data, we can satisfy the non-negativity requirement of distribution for the K-R metric naturally. Moreover, the K-R metric based level-set method can...

We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract...

High-resolution reconstruction of spatial chromosome organizations from chromatin contact maps is highly demanded, but is hindered by extensive pairwise constraints, substantial missing data, and limited resolution and cell-type availabilities. Here, we present FLAMINGO, a computational method that addresses these challenges by compressing inter-de...

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Amp\`{e}re operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the {M}onge-{A}mp{\`e}re eigenvalue problem, {\it Proceedings of the A...

Microwave-induced thermoacoustic imaging (TAI) is a hybrid imaging technique that combines electromagnetic radiation and ultrasonic waves to achieve high imaging contrast and submillimeter spatial resolution. These characteristics make TAI a good candidate to detect material anomalies that change the material electric properties without a noticeabl...

We present Liouville partial-differential-equation (PDE) based methods for computing complex-valued eikonals in the multivalued (or multiple arrival) sense in attenuating media. Since the earth is comprised of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy loss...

We present efficient numerical methods for solving a class of nonlinear Schrödinger equations involving a nonlocal potential. Such a nonlocal potential is governed by Gaussian convolution of the intensity modeling nonlocal mutual interactions among particles. The method extends the Fast Huygens Sweeping Method (FHSM) that we developed in Leung et a...

We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract...

First-arrival traveltime tomography is an essential method for obtaining near-surface velocity models. The adjoint-state first-arrival traveltime tomography is appealing due to its straightforward implementation, low computational cost, and low memory consumption. Because solving the point-source isotropic eikonal equation by either ray tracers or...

We propose a data and knowledge driven approach for SPECT by combining a classical iterative algorithm of SPECT with a convolutional neural network. The classical iterative algorithm, such as ART and ML-EM, is employed to provide the model knowledge of SPECT. A modified U-net is then connected to exploit further features of reconstructed images and...

Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplit...

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator [[EQUATION]] . The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by op...

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The...

We propose a level-set approach for recovering the susceptibility distribution from five-component full magnetic gradient tensor data. Given a value of the average susceptibility, the level-set inversion inverts for the susceptibility distribution so as to depict the shape of the underlying magnetic source. Numerical examples illustrate that the me...

In the present article we extend to the three-dimensional elliptic Monge–Ampère equation the method discussed in Glowinski et al. (J Sci Comput 79:1–47, 2019) for the numerical solution of its two-dimensional variant. As in Glowinski et al. (2019) we take advantage of an equivalent divergence formulation of the Monge–Ampère equation, involving the...

The need to improve the depth resolution of the magnetic susceptibility model recovered from surface magnetic data is a well-known challenge and it becomes increasingly important as the exploration moves to regions under cover and at large depths. Incorporating borehole magnetic data can be an effective means to achieve increased model resolution a...

We propose a Newton-type Gauss–Seidel Lax–Friedrichs sweeping method to solve the generalized eikonal equation arising from wave propagation in a moving fluid. The Lax–Friedrichs numerical Hamiltonian is used in discretization of the generalized eikonal equation. Different from traditional Lax–Friedrichs sweeping algorithms, we design a novel appro...

We propose an L 2 regularized Kantorovich–Rubinstein (KR) metric as a novel misfit function for the level-set based inverse gravity problems without assuming mass conservation. We study the wellposedness of the regularized KR metric and establish the intrinsic relation between the regu- larized KR metric, the KR metric, and the L 2 norm in terms...

We propose a novel Kantorovich-Rubinstein (KR) norm based misfit function to measure the mismatch between gravity-gradient data for the inverse gradiometry problem. Under the assumption that an anomalous mass body has an unknown compact support with a prescribed constant value of density-contrast, we implicitly parameterize the unknown mass body by...

We discuss in this article a novel method for the numerical solution of the two-dimensional elliptic Monge–Ampère equation. Our methodology relies on the combination of a time-discretization by operator-splitting with a mixed finite element based space approximation where one employs the same finite-dimensional spaces to approximate the unknown fun...

The article “A Finite Element/Operator-Splitting Method for the Numerical Solution of the Two Dimensional Elliptic Monge–Ampère Equation”, written by Roland Glowinski, Hao Liu, Shingyu Leung and Jianliang Qian, was originally published electronically on the publisher’s internet portal (currently SpringerLink) on 27 September, 2018 with open access....

This paper is the first attempt to systematically study properties of the effective Hamiltonian $\overline{H}$ arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex $\overline{H}$. Secondly, we a...

Understanding effective Hamiltonians quantitatively is essential for the homogeniza-tion of Hamilton-Jacobi equations. We propose in this article a simple effcient operator-splitting method for computing effective Hamiltonians when the Hamiltonian is either convex or nonconvex in the gradient variable. To speed up our Lie scheme-based operator-spli...

Starting from Hadamard's method, we extend Babich's ansatz to the frequency- domain point-source (FDPS) Maxwell's equations in an inhomogeneous medium in the high-frequency regime. First, we develop a novel asymptotic series, dubbed Hadamard's ansatz, to form the funda- mental solution of the Cauchy problem for the time-domain point-source (TDPS) M...

We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The approach is a natural combination of an asymptotic expansion close to the point source, and an adaptively enriched finite element method based on learning dominant plane wave directions to compute the smooth far fie...

We propose a numerically efficient algorithm for simulating the multi-color optical self-focusing phenomena in nematic liquid crystals. The propagation of the nematicon is modeled by a parabolic wave equation coupled with a nonlinear elliptic partial differential equation governing the angle between the crystal and the direction of propagation. Num...

We have developed a multiple level-set method for inverting magnetic data produced by weak induced magnetization only. The method is designed to deal with a specific class of 3D magnetic inverse problems in which the magnetic susceptibility is known and the objective of the inversion is to find the boundary or geometric shape of the causative bodie...

We present a ray-based finite element method (ray-FEM) by learning basis adaptive to the underlying high-frequency Helmholtz equation in smooth media. Based on the geometric optics ansatz of the wave field, we learn local dominant ray directions by probing the medium using low-frequency waves with the same source. Once local ray directions are extr...

The numerical approximation of high-frequency wave propagation in inhomogeneous media is a challenging problem. In particular, computing high-frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy. In this paper, we propose a new meth...

We have developed a new level-set-based structural parameterization for joint inversion of gravity and traveltime data, so that density contrast and seismic slowness are simultaneously recovered in the inverse problem. Because density contrast and slowness are different model parameters of the same survey domain, we assume that they are similar in...

For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Green's function of th...

Angle Domain Common-Imaging Gathers (ADCIGs) is an important output of pre-stack depth migration. It is the basis of migration velocity analysis, anisotropy analysis and AVA analysis. However, there are still a lot of problems that reduce the efficiency of the existing ADCIGs generating methods and hinder the application of ADCIGs. Methods used to...

We present a ray-based finite element method (ray-FEM) by learning basis adaptive to the underlying high-frequency Helmholtz equation in smooth media. Based on the geometric optics ansatz of the wave field, we learn local dominant ray directions by probing the medium using low-frequency waves with the same source. Once local ray directions are extr...

The usual geometrical-optics expansion of the solution for the Helmholtz equation of a point source in an inhomogeneous medium yields two equations: an eikonal equation for the traveltime function, and a transport equation for the amplitude function. However, two difficulties arise immediately: one is how to initialize the amplitude at the point so...

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation can be viewed as an evolution equation in one of the spatial directions. With such applications in mind, starting from Babich's expansion, we develop a new high-order asymptotic method, which we dub the fast Huygens sw...

We have developed a level-set method for the inverse gravimetry problem of imaging salt structures with density contrast reversal. Under such a circumstance, a part of the salt structure contributes two completely opposite anomalies that counteract with each other, making it unobservable to the gravity data. As a consequence, this amplifies the inh...

Because angle-domain common-image gathers (ADCIGs) from reverse time migration (RTM) are capable of obtaining the correct illumination of a subsurface geologic structure, they provide more reliable information for velocity model building, amplitude-variation versus angle analysis, and attribute interpretation. The approaches for generating ADCIGs m...

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that Maxwell's equations may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green's...

Wave propagation in an isotropic acoustic medium occupied by a moving fluid is governed by an anisotropic eikonal equation. Since this anisotropic eikonal equation is associated with an inhomogeneous Hamiltonian, most of existing anisotropic eikonal solvers are either inapplicable or of unpredictable behavior in convergence. Realizing that this ani...

We propose a novel Babich-like ansatz consisting of an infinite series of dyadic coefficients (three-by-three matrices) and spherical Hankel functions for solving point-source Maxwell's equations in an inhomogeneous medium so as to produce the so-called dyadic Green's function. Using properties of spherical Hankel functions, we derive governing equ...

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation can be viewed as an evolution equation in one of the spatial directions. With such applications in mind, starting from Babich's expansion, we develop new high order asymptotic methods, dubbed the fast sweeping methods,...

Angle-domain common-image gathers (ADCIGs) are important for migration velocity building and AVA analysis et al. The methods used to generating ADCIGs can be mainly divided into direct and indirect methods. These methods all suffer from the degree resolution and computing efficiency. Meanwhile, the migration artifacts occur often if the improper me...

Accurate simulation of seismic waves is of critical importance in a variety of geophysical applications. Based on recent works on staggered discontinuous Galerkin methods, we have developed a new method for the simulations of seismic waves, which has energy conservation and extremely low grid dispersion, so that it naturally provided accurate numer...

We propose an improved fast local level set method for the inverse problem of gravimetry by developing two novel algorithms: one is of linear complexity designed for computing the Frechet derivative of the nonlinear do- main inverse problem, and the other is designed for carrying out numerical continuation rapidly so as to obtain fictitious full me...

We propose a new methodology for carrying out eikonal based traveltime tomography arising from important applications such as seismic imaging and medical imaging. The new method formulates the traveltime tomography problem as a variational problem for a certain cost functional explicitly with respect to both traveltime and sound speed. Furthermore,...

Multiarrival Green's functions are essential in seismic modeling, migration, and inversion. Huygens-Kirchhoff (HK) integrals provide a bridge to integrate locally valid first-arrival Green's functions into a globally valid multiarrival Green's function. We have designed robust and accurate finite-difference methods to compute first-arrival travelti...

We have developed a local level-set method for inverting 3D gravity-gradient data. To alleviate the inherent nonuniqueness of the inverse gradiometry problem, we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain D so that we may convert the o...

We propose a level-set adjoint-state method for crosswell traveltime tomography using both first-arrival transmission and reflection traveltime data. Since our entire formulation is based on solving eikonal and advection equations on finite-difference meshes, our traveltime tomography strategy is carried out without computing rays explicitly. We in...

Motivated by recent theoretical results obtained by the third author for the identification problem arising in single-photon emission computerized tomography (SPECT), we propose an adjoint state method for recovering both the source and the attenuation in the attenuated X-ray transform. Our starting point is the transport-equation characterization...

Motivated by fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods which were originally designed for pure initial-value problems of wave equations, we develop fast multiscale Gaussian beam methods for initial boundary value problems of wave equations in bounded convex domains in the high frequency regime. To compute t...

We propose fast Huygens sweeping methods for Schrödinger equations in the semi-classical regime by incorporating short-time Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) propaga-tors into Huygens' principle. Even though the WKBJ solution is valid only for a short time period due to the occurrence of caustics, Huygens' principle allows us to construct t...

The solution for the eikonal equation with a point-source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source. As such, the eikonal function is not differentiable at the source so that all formally high-order numerical schemes for the eikonal equation yield first-or...

Ptychography is an emerging non-crystalline diffractive imagingtechnique that can potentially reach diffraction limited resolution without theneed for high resolution lenses. To achieve high resolution one must solve a phase-retrieval inverse problem using the diffraction patterns of many partially overlapping sub-image frames. We examine the mathe...

We propose a model for the gravitational field of a floating iceberg D with snow on its top. The inverse problem of interest in geophysics is to find D and snow thickness g on its known (visible) top from remote measurements of derivatives of the gravitational potential. By modifying the Novikov’s orthogonality method we prove uniqueness of recover...

The Gaussian beam method is an asymptotic method for wave equations with highly oscillatory data. In a recent published paper by two of the authors, a multiscale Gaussian beam method was first proposed for wave equations by utilizing the parabolic scaling principle and multiscale Gaussian wavepacket transforms, and numerical examples there demonstr...

In this paper, we obtain the first local a posteriori error estimate for time-dependent Hamilton-Jacobi equations. Given an arbitrary domain Ω and a time T, the estimate gives an upper bound for the L∞-norm in Ω at time T of the difference between the viscosity solution u and any continuous function v in terms of the initial error in the domain of...

The viscosity solution of static Hamilton-Jacobi equations with a point-source condition has an upwind singularity at the source, which makes all formally high-order finite-difference scheme exhibit first-order convergence and relatively large errors. To obtain designed high-order accuracy, one needs to treat this source singularity during computat...

In the geometrical-optics approximation for the Helmholtz equation with a point source, traveltimes and amplitudes have upwind singularities at the point source. Hence, both first-order and higher-order finite-difference solvers exhibit formally at most first-order convergence and relatively large errors. Such singularities can be factored out by f...

The equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop's equilibrium. We propose an adjoint state method to numerically approximate...

We propose a fast local level setmethod for the inverse problemof gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x3-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructiv...

In the high frequency regime, the geometrical-optics approximation for the Helmholtz equation with a point source results in an Eikonal equation for traveltime and a transport equation for amplitude. Because the point-source traveltime field has an upwind singularity at the source point, all formally high-order finite-difference Eikonal solvers exh...

Ptychography promises diffraction limited resolution without the need for high resolution lenses. To achieve high resolution one has to solve the phase problem for many partially overlapping frames. Here we review some of the existing methods for solving ptychographic phase retrieval problem from a numerical analysis point of view, and propose alte...

We present a new algorithm for reconstructing an unknown source in Thermoacoustic and Photoacoustic Tomography based on the recent advances in understanding the theoretical nature of the problem. We work with variable sound speeds that might be also discontinuous across some surface. The latter problem arises in brain imaging. The new algorithm is...

We present an efficient algorithm for reconstructing an unknown source in thermoacoustic and photoacoustic tomography based on the recent advances in understanding the theoretical nature of the problem. We work with variable sound speeds that also might be discontinuous across some surface. The latter problem arises in brain imaging. The algorithmi...

In this work an adaptive strategy for the phase space method [5] for traveltime tomography is developed. The method first uses those geodesics/rays that produce smaller mismatch with the measurements and continues on in the spirit of layer stripping without defining the layers explicitly. The adaptive approach improves stability, efficiency and acc...

We propose the backward phase flow method to implement the Fourier–Bros–Iagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schrödinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in [12]. In this paper we aim at two crucial computational issues of the Eulerian Gauss...

This paper introduces a wavepacket-transform-based Gaussian beam method for solving the Schrödinger equation. We focus on addressing two computational issues of the Gaussian beam method: how to generate a Gaussian beam representation for general initial conditions and how to perform long time propagation for any finite period of time. To address th...

We propose an effective stopping criterion for higher-order fast sweeping schemes for static Hamilton-Jacobi equations based on ratios of three consecutive iterations. To design the new stopping criterion we analyze the convergence of the first-order Lax-Friedrichs sweeping scheme by using the theory of nonlinear iteration. In addition, we propose...

We prove optimal convergence rates for the approximation provided by the original discontinuous Galerkin method for the transport-reaction problem. This is achieved in any dimension on meshes related in a suitable way to the possibly variable velocity carrying out the transport. Thus, if the method uses polynomials of degree $k$, the $L^2$-norm of...

We introduce a new multiscale Gaussian beam method for the numerical solution of the wave equation with smooth variable coefficients. The first computational question addressed in this paper is how to generate a Gaussian beam representation from general initial conditions for the wave equation. We propose fast multiscale Gaussian wavepacket transfo...

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrödinger equation. Traditional Gaussian beam type methods for the Schrödinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian...

We propose a new sweeping algorithm which utilizes the Legendre transform of the Hamiltonian on triangulated meshes. The algorithm is a general extension of the previous proposed algorithm by Kao et al. [C.Y. Kao, S.J. Osher, Y.-H. Tsai, Fast sweeping method for static Hamilton–Jacobi equations, SIAM J. Numer. Anal. 42 (2005) 2612–2632]. The algori...

We adopt a recent work in Chung, Qian, Uhlmann and Zhao (Inverse Problems, 23(2007) 309-329) to develop a phase space method for reconstructing pressure wave speed and shear wave speed of an elastic medium from travel time measurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannian metrics with their trav...

We develop high order essentially non-oscillatory (ENO) schemes on non-uniform meshes based on generalized binary trees. The idea is to adopt an appropriate data structure which allows to communicate information easily between unstructured data structure and virtual uniform meshes. While the generalized binary trees as an unstructured data structu...

We design an Eulerian Gaussian beam summation method for solving Helmholtz equations in the high-frequency regime. The traditional Gaussian beam summation method is based on Lagrangian ray tracing and local ray-centered coordinates. We propose a new Eulerian formulation of Gaussian beam theory which adopts global Cartesian coordinates, level sets,...

Gaussian beams are approximate solutions to hyperbolic partial differential equations that are concentrated on a curve in space-time. In this paper, we present a method for computing the stationary in time wave field that results from steady air flow over topography as a superposition of Gaussian beams. We derive the system of equations that govern...

A new numerical procedure coupling the level-set method with the moving-mesh method to simulate subcooled nucleate pool boiling is proposed. Numerical test problems have vali- dated this new method. The simulation of bubble dynamics during nucleate boiling under liquid subcooling shows that this novel adaptive method is more accurate in determining...

We develop a fast sweeping method for static Hamilton-Jacobi equations with con- vex Hamiltonians. Local solvers and fast sweeping strategies apply to structured and unstructured meshes. With causality correctly enforced during sweepings numerical evidence indicates that the fast sweeping method converges in a nite number of iter- ations independen...

We propose a new formulation for traveltime tomography based on paraxial Liouville equations and level set formulations. This new formulation allows us to account for multivalued traveltimes and multipathing systematically in the tomography problem. To obtain efficient implementations, we use the adjoint state technique and the method of gradient d...

We develop a new phase space method for reconstructing the index of refraction of a medium from travel time measurements. The method is based on the so-called Stefanov–Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The new algorithm is a...

The original fast sweeping method, which is an efficient iterative method for stationary Hamilton-Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple refere...

We design a class of Weighted Power-ENO (Essentially Non-Oscillatory) schemes to approximate the viscosity solutions of Hamilton-Jacobi (HJ) equations. The essential idea of the Power-ENO scheme is to use a class of extended limiters to replace the minmod type limiters in the classical third-order ENO schemes so as to improve resolution near kinks...

We construct high order fast sweeping numerical methods for computing viscosity solu- tions of static Hamilton-Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximation to derivatives, monotone numerical Hamiltonians and Gauss Seidel iterations with alternating-direction sweepi...

We set up the electromagnetic system and its plane-wave solutions with the associated slowness and wave surfaces. We treat the Cauchy initial-value problem for the electric vector and make explicit the quantities necessary for numerical evaluation. We use the Herglotz-Petrovskii representation as an integral around loops which, for each position an...

We propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two-dimensional (2-D) acoustic wave equations. The geometrical optics term consists of two multivalued functions: a travel-time function satisfying the eikonal equation locally and an amplitude function...

RESUMEN RESUMEN Anew monotone finite difference scheme is introduced that approximates viscositysolutions of first-order nonlinear Hamilton--Jacobi equations. The main feature ofthe scheme is that it allows for locally varying timeand space grids , making it ideal for use with adaptivealg...